LMFDB - Embedded newform 9280.2.a.w.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-4,0,-2,0,4,0,2,0,-4,0,4,0,4,0,0,0,-4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} -1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +0.828427 q^{11} +2.00000 q^{13} +2.00000 q^{15} +2.82843 q^{17} -4.82843 q^{19} -9.65685 q^{21} +3.17157 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} -6.48528 q^{31} -1.65685 q^{33} -4.82843 q^{35} +8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +11.6569 q^{47} +16.3137 q^{49} -5.65685 q^{51} +3.65685 q^{53} -0.828427 q^{55} +9.65685 q^{57} +3.65685 q^{61} +4.82843 q^{63} -2.00000 q^{65} +6.48528 q^{67} -6.34315 q^{69} +15.3137 q^{71} +8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} +2.48528 q^{79} -11.0000 q^{81} +7.17157 q^{83} -2.82843 q^{85} +2.00000 q^{87} -7.65685 q^{89} +9.65685 q^{91} +12.9706 q^{93} +4.82843 q^{95} -12.4853 q^{97} +0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^23 + 2 * q^25 + 8 * q^27 - 2 * q^29 + 4 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 12 * q^41 - 12 * q^43 - 2 * q^45 + 12 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 8 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^77 - 12 * q^79 - 22 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^91 - 8 * q^93 + 4 * q^95 - 8 * q^97 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	−9.65685	−2.10730
$22$	0	0
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.48528	−1.16479	−0.582395	−	0.812906i	$-0.697884\pi$
$31$	−6.48528	−1.16479	−0.582395	+	0.812906i	$0.697884\pi$
$32$	0	0
$33$	−1.65685	−0.288421
$34$	0	0
$35$	−4.82843	−0.816153
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	−5.65685	−0.792118
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	9.65685	1.27908
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	3.65685	0.468212	0.234106	−	0.972211i	$-0.424784\pi$
$61$	3.65685	0.468212	0.234106	+	0.972211i	$0.424784\pi$
$62$	0	0
$63$	4.82843	0.608325
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	6.48528	0.792303	0.396152	−	0.918185i	$-0.370345\pi$
$67$	6.48528	0.792303	0.396152	+	0.918185i	$0.370345\pi$
$68$	0	0
$69$	−6.34315	−0.763625
$70$	0	0
$71$	15.3137	1.81740	0.908701	−	0.417447i	$-0.137075\pi$
$71$	15.3137	1.81740	0.908701	+	0.417447i	$0.137075\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	2.48528	0.279616	0.139808	−	0.990179i	$-0.455351\pi$
$79$	2.48528	0.279616	0.139808	+	0.990179i	$0.455351\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	7.17157	0.787182	0.393591	−	0.919286i	$-0.371233\pi$
$83$	7.17157	0.787182	0.393591	+	0.919286i	$0.371233\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−7.65685	−0.811625	−0.405812	−	0.913956i	$-0.633011\pi$
$89$	−7.65685	−0.811625	−0.405812	+	0.913956i	$0.633011\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	0	0
$93$	12.9706	1.34498
$94$	0	0
$95$	4.82843	0.495386
$96$	0	0
$97$	−12.4853	−1.26769	−0.633844	−	0.773461i	$-0.718524\pi$
$97$	−12.4853	−1.26769	−0.633844	+	0.773461i	$0.718524\pi$
$98$	0	0
$99$	0.828427	0.0832601

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.w.1.2		2
4.3	odd	2		9280.2.a.be.1.1		2
8.3	odd	2		145.2.a.b.1.2	✓	2
8.5	even	2		2320.2.a.k.1.2		2
24.11	even	2		1305.2.a.n.1.1		2
40.3	even	4		725.2.b.c.349.2		4
40.19	odd	2		725.2.a.c.1.1		2
40.27	even	4		725.2.b.c.349.3		4
56.27	even	2		7105.2.a.e.1.2		2
120.59	even	2		6525.2.a.p.1.2		2
232.115	odd	2		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	8.3	odd	2
725.2.a.c.1.1		2	40.19	odd	2
725.2.b.c.349.2		4	40.3	even	4
725.2.b.c.349.3		4	40.27	even	4
1305.2.a.n.1.1		2	24.11	even	2
2320.2.a.k.1.2		2	8.5	even	2
4205.2.a.d.1.1		2	232.115	odd	2
6525.2.a.p.1.2		2	120.59	even	2
7105.2.a.e.1.2		2	56.27	even	2
9280.2.a.w.1.2		2	1.1	even	1	trivial
9280.2.a.be.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} -1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +0.828427 q^{11} +2.00000 q^{13} +2.00000 q^{15} +2.82843 q^{17} -4.82843 q^{19} -9.65685 q^{21} +3.17157 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} -6.48528 q^{31} -1.65685 q^{33} -4.82843 q^{35} +8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +11.6569 q^{47} +16.3137 q^{49} -5.65685 q^{51} +3.65685 q^{53} -0.828427 q^{55} +9.65685 q^{57} +3.65685 q^{61} +4.82843 q^{63} -2.00000 q^{65} +6.48528 q^{67} -6.34315 q^{69} +15.3137 q^{71} +8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} +2.48528 q^{79} -11.0000 q^{81} +7.17157 q^{83} -2.82843 q^{85} +2.00000 q^{87} -7.65685 q^{89} +9.65685 q^{91} +12.9706 q^{93} +4.82843 q^{95} -12.4853 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^23 + 2 * q^25 + 8 * q^27 - 2 * q^29 + 4 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 12 * q^41 - 12 * q^43 - 2 * q^45 + 12 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 8 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^77 - 12 * q^79 - 22 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^91 - 8 * q^93 + 4 * q^95 - 8 * q^97 - 4 * q^99

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